On the affinization of a nilpotent orbit cover

Let $\mathfrak{g}$ be a simple classical Lie algebra over $\mathbb{C}$ and $G$ be the adjoint group. Consider a nilpotent element $e\in \mathfrak{g}$, and the adjoint orbit $\mathbb{O}=Ge$. The formal slices to the codimension $2$ orbits in the closure $\bar{\mathbb{O}}\subset \mathfrak{g}$ are well-known due to the work of Kraft and Procesi. In this paper, we prove a similar result for the universal $G$-equivariant cover $\tilde{\mathbb{O}}$ of $\mathbb{O}$. Namely, we describe the codimension $2$ singularities for its affinization $Spec(\mathbb{C}[\tilde{\mathbb{O}}])$.